To find the degree of the polynomial that produces the given successive results, we analyze the sequence of differences until we find a constant difference. Here are the successive results: 24, 0, -24, 0, 144, 504.
Let's go through the steps to find the differences:
1. First differences:
[tex]\[
0 - 24 = -24
\][/tex]
[tex]\[
-24 - 0 = -24
\][/tex]
[tex]\[
0 - (-24) = 24
\][/tex]
[tex]\[
144 - 0 = 144
\][/tex]
[tex]\[
504 - 144 = 360
\][/tex]
So, the first differences are: -24, -24, 24, 144, 360.
2. Second differences:
[tex]\[
-24 - (-24) = 0
\][/tex]
[tex]\[
24 - (-24) = 48
\][/tex]
[tex]\[
144 - 24 = 120
\][/tex]
[tex]\[
360 - 144 = 216
\][/tex]
So, the second differences are: 0, 48, 120, 216.
3. Third differences:
[tex]\[
48 - 0 = 48
\][/tex]
[tex]\[
120 - 48 = 72
\][/tex]
[tex]\[
216 - 120 = 96
\][/tex]
So, the third differences are: 48, 72, 96.
4. Fourth differences:
[tex]\[
72 - 48 = 24
\][/tex]
[tex]\[
96 - 72 = 24
\][/tex]
So, the fourth differences are: 24, 24, which is constant.
Since it took four levels of differences to reach a constant difference, the degree of the polynomial is 4.
Therefore, the correct answer is:
c) 4
